14.6 Project Archimedes' Floating Paraboloid

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The Volume and Centroid of the Segment The vertex V of the segment is the point of the parabola z = y 2 in the yz-plane x = 0 where the tangent line is parallel to the major axis P1P2 of the top base. The slope of this line is m = (P2[[3]]-P1[[3]]) / (P2[[2]]-P1[[2]]); so the y-coordinate of V is given by yv = First[y /. Solve[2*y == m, y]]; Hence the vertex is V = {0, yv, yv^2}; The normal vector vector to the base plane z = y + 2 is n = {0, -1, 1}; and the height h of the inscribed cone is the component of VM = M - V; in the direction of n. The familiar dot product formula therefore gives h = n . VM/Sqrt[n . n] 9 4 2 Finally, Archimedes' theorem that the volume of the segment is 3/2 times the volume of the inscribed elliptical cone with the same vertex V gives volume = (3/2)*(1/3)*A*h 81π 81 Pi / 32 32 This is the same volume we found by multiple integration. And Archimedes' centroid computation also agrees with our multiple integral computations, because the point on VM that is 2/3 of the way from V to M is centroid = (2/3)M + (1/3)V {0, 1/2, 7/4} 14.6 Project 12
Using MATLAB Taking a = 1 and b = 2 for an example, we want to investigate the segment of a paraboloid that is bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = y + 2. If we view this oblique segment looking inward along the x-axis, we see that it is bounded front-and-back by the surfaces defined on the yz-region that is bounded above by the line z = y + 2 and below by the parabola z = y 2 . This region lies over the interval –1 < y < 2 on the y-axis, as illustrated in the following figure. 14.6 Project 13
Page 1 and 2: 14.6 Project Archimedes' Floating P
Page 3 and 4: z = x 2 + y 2 and below by the plan
Page 5 and 6: Using Maple P1 y M 14.6 Project 5 C
Page 7 and 8: Thus our oblique segment of a parab
Page 9 and 10: centroid = (2/3)*M + (1/3)*V; Using
Page 11: Archimedes' Way First we examine he
Page 15 and 16: Archimedes' Way First we examine he

14.6 Project Archimedes' Floating Paraboloid

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